Endgame analysis for Hexagonal Chess

Dave McCooey did an extensive computer analysis of all endgames with two
pieces (and two kings) on a hexagonal board, without pawns, with the pieces
taken from the standard pieces and a collection of 9 fairy pieces.
The board is the 91 square hexagonal board from McCooey's Hexagonal Chess, which is also used in Glinski's Hexagonal Chess. As the moves of the pieces
in these two variants are the same except for pawns, the results are valid for
both hexagonal chess variants.

Moves like a Knight except that it may continue moving in a
straight line through unoccupied squares, each a Knight's
move from the last and in the same direction away from the starting square.

Duke (symbol = D)

Moves like a King or a Knight

Camel (symbol = L)

Moves to any of the 12 hexes that can be reached by moving
3 hexes Rook-wise, making a 60-degree turn, and then moving
1 hex Rook-wise. It hops from hex to hex like a Knight.
(Note that, unlike its square-chess counterpart, a Hex Chess
Camel is not restricted to a single color.)

Zebra (symbol = Z)

Moves to any of the 12 hexes that can be reached by moving
3 hexes Rook-wise, making a 60-degree turn, and then moving
2 hexes Rook-wise. It hops from hex to hex like a Knight.

Wazir (symbol = W)

Moves like a Rook, but only one square at a time.

Ferz (symbol = F)

Moves like a Bishop, but only one square at a time.

Whenever an endgame involves at least one piece that is restricted to a
single color (e.g. Bishop or Ferz), then multiple summaries are reported
- one for each combination of the colors and/or hexagonal lattices of the
those pieces which are colorbound. Such summaries are indicated by a
parenthesized 2-letter suffix. In the following definitions, the term
"central" means a colorbound piece that is of the same color as the
center hex, and the term "non-central" means a colorbound piece whose
color doesn't match that of the center hex.

(dn)

Both pieces are non-central, and they are of different colors.

(de)

Both pieces are on the same side, exactly one of them is central,
and the other is non-central.

(cn)

The 1st piece is central, and the 2nd piece is non-central.

(nc)

The 2nd piece is central, and the 1st piece is non-central.

(sn)

Both pieces are non-central, and they are of the same color.

(cc)

Both pieces are central.

(cx)

The 1st piece is central, and the 2nd piece is NOT colorbound.

(nx)

The 1st piece is non-central, and the 2nd piece is NOT colorbound.

(xc)

The 2nd piece is central, and the 1st piece is NOT colorbound.

(xn)

The 2nd piece is non-central, and the 1st piece is NOT colorbound.

Definitions used in the statistics

Unique Legal Positions:

Each legal position is one of either 6 or 12 equivalent positions
due to rotations and/or reflections. If all pieces are on either
a certain central diagonal or rank, then the number is 6. Otherwise
it is 12. The Unique Legal Positions considers a set of equivalent
positions as a single position.

WHITE win:

WHITE can force a win.

BLACK win:

BLACK can force a win.

Simple draw:

The weaker side can force a simplification of the position
(i.e. by capture), resulting in a simplified drawn position.

Perpetual check:

The weaker side cannot force a simplification but can nevertheless
force a sequence of positions in which the stronger side cannot
escape being checked without either losing or allowing simplification.

Fortress draw:

The weaker side can force neither a simplification nor a perpetual
check but can nevertheless hold the draw.

Longest win:

The maximum number of half-moves that it can require to force
checkmate or to force a conversion (via capture) to a simpler ending
that is also won. Note that a conversion to a simpler ending can
result from the winning side making the capture or the losing side
making the capture. Thus, there are some (but not many) endings
where the longest wins are positions where the winning side's
quickest course is to force the losing side to capture one of the
winning side's pieces, preserving the win, of course.

Checkmate

A type of win where the quickest course for the winning side is
to checkmate the losing side.

Forced Capture

A type of win where the quickest course for the winning side is
to force the losing side to capture one of the winning side's pieces,
preserving the win.

Capture

A type of win where the quickest course for the winning side is
to capture one of the losing side's pieces.

Central

A colorbound piece that is restricted to the same color hexes as
the center hex.

Non-central

A colorbound piece that is restricted to hexes of a color different
than the center hex.

The string used to describe an endgame shows the WHITE side first,
then the letter 'v', then the BLACK side. Thus, KRvKN means WHITE
King+Rook versus BLACK King+Knight. In each endgame, there is a
strong side and a weak side, according to which side has more total
wins. In all of these endgames, the strong side was chosen to be WHITE.

The endgames are divided into the following sub-types:

strong side has 2 pieces, weak side has 0 pieces

strong side has 1 pieces, weak side has 1 piece

There were 6 endgames where the longest win requires 100 or more
half-moves, and, in all cases, it was the strong (WHITE) side to win.

The Queen and the Chancellor are about equal strength.
The Queen is better against an Amazon or a Pegasus, but
the Chancellor is much better against a Rook.

A Queen does not beat a Rook in general.

A Queen does not beat a Duke in general.

A Queen beats any piece from the set {G,I,N,L,Z,W,B,F}.

A Chancellor comes very close to beating a Rook.

A Chancellor does not beat a Duke in general.

A Chancellor does not beat a Pegasus in general.

A Chancellor beats any piece from the set {I,N,L,Z,W,B,F}.

A Rook is better than a Duke, but the KRvKD endgame is a draw
in general.

A Rook does not beat a Pegasus in general.

A Rook does not beat a Knightrider in general.

A Rook beats any piece from the set {N,L,Z,W,B,F}.

A Rook can checkmate a long King.

A Duke does not beat a Pegasus in general.

A Duke does not beat a Knightrider in general.

A Duke beats any piece from the set {N,L,Z,W,B,F}.

A Duke can checkmate a long King.

The Pegasus is much weaker than its square-chess counterpart.
It cannot even force checkmate against a long King. It is
measurably weaker than both the Rook and Duke, although
both the KRvKG and KDvKG endgames are draws in general.

A Pegasus and any piece from the set {G,I,N,L,Z,W,B,F} can
checkmate a lone King.

A Knightrider and any piece from the set {I,N,L,Z,W} can
checkmate a lone King.

A Knightrider and a Bishop cannot checkmate a lone King.

A Knight and any piece from the set {N,L,Z,W} can
checkmate a lone King.

A Knight and a Bishop cannot checkmate a lone King.

A Camel and a Wazir can checkmate a lone King.

A Zebra and a Wazir can checkmate a lone King.

In many cases, two Wazirs can checkmate a lone King, but
there are a large number (16.6%) of fortress draws.

A Wazir and a Ferz can checkmate a lone King, but only
if the Ferz is NOT of the same color as the center hex.

A Wazir and a Bishop can checkmate a lone King.

In 2-piece endgames, a Bishop is a very weak piece, even
weaker than a Wazir.